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Improving bending stress in spur gears using asymmetric gears and

shape optimization

Niels L. Pedersen ⁎

Dept. of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, Nils Koppels Allé, Building 404, DK-2800 Kgs. Lyngby, Denmark

a r t i c l e i n f o a b s t r a c t

Article history:

Received 17 July 2009

Received in revised form 1 June 2010

Accepted 8 June 2010

Available online 23 July 2010

Bending stress plays a significant role in gear design wherein its magnitude is controlled by the

nominal bending stress andthe stress concentration due to the geometrical shape.The bending

stress is indirectly related to shape changes made to the cutting tool. This work shows that the

bending stress can be reduced significantly by using asymmetric gear teeth and by shape

optimizing the gear through changes made to the tool geometry. However, to obtain the largest

possible stress reduction a custom tool must be designed depending on the number of teeth,

but the stress reductions found are not very sensitive to small design changes. This observation

suggests the use of two new standard cutting tools.

© 2010 Elsevier Ltd. All rights reserved.

Keywords:

Gear

Asymmetry

Bending stress

Tool design

External spur

FEM

1. Introduction

Gear strength is influenced by geometry, the subject of this paper, as well as by material selection and production processes.

Two primary fatigue related failure modes determine gear strength; failure due to bending stress and failure due to contact

pressure. The latter failure is primarily due to pitting while the former is due to tooth breakage. The focus of this work is on

reducing bending stress levels whereby improving gear strength.

Gear design is in most cases conservative and specified by different standards. Almost all gears exhibit involute shape because

the contact forces act along a straight line and a center distance variation due to, e.g. manufacturing tolerances or loadings, does

not influence this fact. A center variation will neither influence the gear ratio. The only design variable that controls the involute

shape is the pressure angle α ; it is typically assigned the value α =20°. Only the gear region that is in contact with the other gear in

the mesh is described by the involute shape. The root geometry or bottom land region, that connects two neighboring teeth, can be

designed rather freely. The task of this paper, also done in Ref. [1], is to improve the gear strength by changing the gear geometry in

a way that retains the involute shape.Design changes of the gears are achieved indirectly by redesigning the cutting tool. Cutting tool parameterization includes the

possibility of an asymmetric tooth; it is simple as it only requires four design parameters. Resulting optimized designs show that a

significant reduction in the bending stress is possible. Furthermore, the cutting tool shape is described analytically and hence so is

the cut teeth shape Thus, as indicated in Ref. [1], the maximum stress calculations can be trusted.

The following Section 2 discusses the general aspects of gear design with a special focus on the asymmetric design and relates

the present work to existing works. A discussion of the herein used finite element analysis is also provided. Geometric

parameterization of the cutting tool and the analytical shape of a gear cut with this are resented in Sections 3 and 4. Section 5

Mechanism and Machine Theory 45 (2010) 1707–1720

⁎ Tel.: +45 45255667; fax: +45 45254250.

E-mail address: [email protected].

0094-114X/$–

see front matter © 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechmachtheory.2010.06.004

Contents lists available at ScienceDirect

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http://dx.doi.org/10.1016/j.mechmachtheory.2010.06.004



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presents optimized designs of spur gears with different number of teeth. These results lead to the two new suggested standard

cutting tools presented in Section 6.

2. General aspects of gear design and analysis

Contact zone geometry is totally controlled by the involute shape and the only way to reduce the contact stress (Hertzian) is to

increase the pressure angle, when keeping the width constant. Pressure angle increase will increase the force on the surface under an

assumption of constant transmitted torque, however, the curvatureradius is increased so altogether a reductionin the surface stress is

the result. A disadvantage of increasing the pressure angle is a reduction in the contact ratio. With standard ISO teeth the bending

stress is also reducedwith increasingpressureangle. Again theload onthe teeth is increasedbut at thesame time the root width of the

teeth is increased,so in total a reductionin the bending stressis theresult. With a pressure angle decrease, theprimary advantage is the

increase of the contact ratio and a possible increase of the tooth height that will have a positive influence on the noise level. A

disadvantage is an increase in the maximum teeth sliding speed that has a negative influence on the lubrication. A further listing of

advantages and disadvantages of increasing or decreasing the pressure angle can be found in Ref. [2].

Almost all gears are symmetric and defined according to the standard cutting tool. The cutting tool definition used in the

present paper is based on the ISO profile and seen in Fig. 1.

The shown profile has, as the ISO profile, an added top with the height of M /4, where M is the gear module which controls the

gear teeth size and subsequent also the gear size (the pitch diameter d p is given by d p= Mz where z is the number of teeth on the

gear). Top radius ρ is chosen such that there is no jump in the slope. The bottom of the true cutting tooth profile is not identical to

the bottom of the shown cutting profile based on the ISO profile. For the real cutting profile, the top of the cut teeth is assumed

given by the initial steel blank diameter, which is equal to the addendum diameter. The shown profile has as envelope the full cut

tooth, i.e., the envelope of the bottom part of the profile is the finished cut tooth top. Teeth cut with the ISO profile and teeth cut

with the profile shown in Fig. 1 are therefore identical.

Gears become symmetric when the cutting tool tooth is symmetric with respect to the y-axis as defined in Fig. 1. Symmetry of

the twoinvolutes of a tooth follows a choice of identical pressure angles, α d= α c . Subscript d is used for drive side and subscript c is

used for coast side. Two identical pressures angles imply that the two straight lines have opposite gradient, and that they go

through the points (−π M /4, 0) and (π M /4, 0) respectively (the envelope of the straight side are the tooth involute).

If the object of gear design is to minimize the stresses, it follows from the listed advantages and disadvantages of changing the

pressure angle that the pressure angle should be as large as possible. The limiting factor is the needed minimum tooth top thickness.

In many applications the gears only work in a unidirectional way, or not with the same loading conditions in the two opposite

directions. From this follows that it might be an advantage to use different pressure angle values, i.e., an asymmetric gear design.

This idea is not new and can be found in, e.g. Refs. [3,4], the two papers also mention earlier work done in the Soviet Union. The two

papers are short (2 pages) and the information primarily relates to basic geometry. In the recent years the subject has gained an

increasing interest this can be seen from the number of papers published. The papers, Refs. [5–9] show that the use of asymmetric

gear teeth has a big potential.

More information with regard to design of asymmetric gear teeth can be found in e.g., Refs. [10–20]. The present paper differs

from these papers by the way the shape optimization is performed. It can be dif ficult to see directly what is gained by using

asymmetric gears. Primarily this is due to the nature of gear design; it has many variables and these can be varied in many ways. In

Ref. [19] the results show that the best design with respect to bending stress is the symmetric design but this paper kept the

Fig. 1. Cutting profile geometric definition and the basic profile based on the ISO profile, M is the gear module that defines the teeth size in the gear. The two sidesof the tool are termed drive and coast side respectively. Pressure angles α d and α c are shown here with the same value. The coordinate system used is shown.

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pressure angle at the input or drive side to the standard α d=20° and limits the pressure angle on the opposite or coast side to be

α c ≦20°. If we allow for a larger coast side pressure angle the bending stress becomes smaller as shown in the present paper.

In many papers on asymmetric tooth design, the cutting tool top is a circle, see e.g. Ref. [18]. With this choice of design, the full

optimization with respect to bending stress is not found. The positive effect on the bending stress is then solely related to a root

thickness increase of the tooth, but the stress can be further reduced using shape optimization to reduce the stress concentration.

In the papers on asymmetric tooth design there seems to be two choices: either α dbα c , i.e., the drive side pressure angle is

smaller than the coast side pressure angle or α dNα c . In Ref. [10] the choice made is α dNα c . It is stated that this choice is made

because it reduces the mesh stiffness that has a positive influence on noise and vibration levels. In Ref. [14] the choice made is

α dbα c and here it is shown that this only has a little influence on the mesh stiffness. The present paper does not discuss mesh

stiffness and both choices of relative pressure angles are shown in the examples in order to explore the possible advantages.

When increasing pressure angles the negative result is that the top land thickness becomes smaller and in the limit it becomes

pointed, with a further increase in the pressure angle the teeth become shorter. A shorter teeth decrease the contact ratio, which is

undesirable. In Ref. [21] the minimum top land thickness limit is sa≧0.25 M , for carburized gears the limit is reported to be higher

sa≧0.4 M .

Shape optimization of gear teeth can be done in two principal different ways. The direct and most simple way is to optimize the

tooth root directly as it is done in e.g. Refs. [6,22]. Another way of optimizing is changing the shape of the tool that cuts the shape.

The latter method is used here, the result being that the tool becomes a custom tool for the specific gear (number of teeth). This

method of optimization was also used in Ref. [1], but the method is in the present paper extended to include asymmetric teeth

design. In the optimization the focus is on the bending stress because this is the only limiting stress that is affected by the design

changes made in the bottom land. Surface stresses are only affected, as stated previously, indirectly through the choice of pressure

angle. In the different optimizations presented in the present paper, the pressure angles are selected within limits controlled by

the top land thickness and as such, the surface stresses are not the optimization objective.

From shape optimization, see e.g. Ref. [23], we know that the shape parameterization is of great importance and the node

positions in the finite element parameterization should not be used as design parameters directly. Instead the cutting tool tip is

analytically parameterized using a super elliptical shape. It is also known from shape optimization for minimum stress

concentration that the primary first design criterion is that the slope is continuous, i.e., a corner may imply infinite stress.

2.1. FE modeling

An assumption of plane stress is made in the present paper since the optimizations are made for external spur gears. The

Poisson's ratio used is ν =0.3 and linear elasticity is assumed. The geometry, the loads and the supports need to be specified for the

FE modeling. In Fig. 2 a schematic drawing of a rack tooth is presented.

The geometry is given by the cuttingtool design, see Section 3. The load is acting perpendicular to the surface. In realitythe load

does not act as a single load as indicated in Fig.2, but is spread out as given by the Hertzian pressure distribution and could be used

in the FE simulations. We apply a constant line load that is perpendicular to the surface because this study is related to the root

stress and not to the contact stress. The difference in the maximum bending stress from applying a constant line force or a force

that varies according to Hertz is negligible. Actual load size is not important because of the assumption of linear elasticity.

Support placements are a compromise. The tooth is fixed at a depth of size M and at two symmetry lines to the two adjacent

teeth, see Fig. 2. If the support is moved to the tooth root it would comply with the Lewis formula for bending stress calculation, see

e.g. Ref. [24]. By moving the support closer to the gear center the tooth becomes more flexible. Fixing the tooth at a depth larger

than M has a negligible influence on the stress we want to minimize, i.e., the maximum bending stress at the root.

Fig. 2. Schematic drawing of one rack tooth.

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3. Geometry parameterization

The basic ISO profile has an added top with a height of M /4 as seen in Fig. 1. The top on the basic profile is added to make a

clearance for the lubricating oil. It is however also the top that controls the bottom shape of the gear teeth. This is the case which

can be seen e.g. for the rack where the teeth are the counter part of the profileas shown in Fig. 1. In Ref. [1] the tool tip is the design

domain because the symmetric design was kept and the focus was on the stress concentrations at the tooth root. Root stress is

directly controlled by the tool tip shape.

In the present paper the parameterization used in Ref. [1] is extended to include asymmetric cutting teeth that give asymmetric gear

teeth. The straight side of the cutting tool is no longer fixed to have the pressure angle α but the idea is to change this value without

interfering with a lower constraint on the tooth top thickness(top land). This leaves most of the parameterization to the tool top.

In order to find the limits to the pressure angles a relation between the pressure angles and the top land thickness is needed. A

derivation is given in Ref. [10] and an alternative derivation is given here. Since the lengths are different on the two sides we

distinguish between drive side and coast side. Tooth thickness at the pitch diameter is given by

s = s pd + s pc ð1Þ

where s pd and s pc are the tooth pitch thickness at the pitch diameter for the drive and coast side, respectively. These are given by

s pd =1

4π + pstanðαdÞ

M ð2Þ

s pc =1

4π + pstanðαc Þ M ð3Þ

where ps is a possible cutting tool shift. This shiftis in many cases not identical onthe two mating gears. The lengths relatedto thedrive

side are indicated in Fig. 3 where we have used the known involute function definition given by inv(α )=tan(α )−α .

In Fig. 3 only the part that relates to the coast side is shown for clarity. A similar figure can be made for the drive side, the only

difference is that the base diameter is different, in the shown case the base radius for the drive side r bd is smaller than the base

radius for the coast side r bc .

At the top diameter dt = 2r t the tooth thickness for the drive and coast side is given by

std =dt

2

s

d p

+ invðαdÞ−invðαtdÞ

!ð4Þ

stc =

dt

2

s

d p + invðαc Þ−invðαtc ÞÞ !

ð5Þ

Fig. 3. Illustration of the tooth topthickness foran asymmetric tooth related to only one side, here the coast side. Thecoast side pressure angle is here α c =20° andthe drive side pressure angle is α d=30° for a gear with 17 teeth.



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It should be noted that one of the length std or stc might be negative, indicating that all ofthe tooth top lies to one siderelative to the

center line, see Fig. 3. Total tooth top length is given directly by

st = std + stc ð6Þ

It remains to determine the drive top angle, α td, and the coast top angle, α tc , these are given by

cosðαtdÞ =d p

dt cosðαdÞ ð7Þ

cosðαtc Þ =d p

dt

cosðαc Þ ð8Þ

From the input:

1. Module M .

2. Shift values for the two gears ps1 and ps2.

3. Number of teeth on the two gears z1 and z2.

4. A given pressure angle, α d or α c .

5. A lower limit on the tooth top thickness, e.g., st =0.25 M .

The limiting value of the other pressure angle can be found by the use of Eqs. (1)–

(8) and, e.g., Newton–

Raphson iterations. Inthe examples shown in the present paper either the drive pressure angle or the coast pressure angle is set to the standard value 20°

while the other pressure angle is changed.

The parameterization that remains is the tool top, the number of possible tool top design parameterizations is infinite. Here a

variation of the parameterization used in Ref. [1] is applied, the central part is to use a variation of the super ellipse. Focus is on

simplicity, although the optimization result should still be near to the optimal design. That a given parameterization is suf ficiently

flexible, i.e. that it can return optimal designs, can only be checked or verified after an actual optimization procedure. If the stress is

constant along major parts of the surface then the shape is assumed optimal, see e.g. Ref. [25].

The parameterization presented fulfills the following constraints:

• The added tool tip height is fixed at M /4.

• The involute part of the tooth must not be penetrated on the drive side.

That the tool tip height is kept fixed is applied in order to allow for the same clearance in the optimized gears as is the case for

the ISO gears. The involute part shouldbe kept unchanged to allow the optimized gears to have the same functional qualities as theoriginal involute gears.

A distinction is made between the tool top part that cuts the tooth root of the drive side (drive top) and the other part that cuts

the tooth root of the coast side (coast top). As indicated in Fig. 4 the coast side top is a simple circle (part of a full circle).

The radius of the circle is given as

ρc = κM =4μ + π−5tanðαc Þ

4ðcosðαc Þ−ðsinðαc Þ−1Þtanðαc ÞÞM ð9Þ

and might be greater than or smaller than the ISO standard ρ≈0.38 M (see Fig. 1). This also means that the involute on the costs

side might not be as long as it would have been using the ISO cutting tooth, but this is ignored because of the unidirectional loading

assumption.

Fig. 4. The design domain for the optimization shown as the hatched part. The coast side pressureangle and drive side pressure angle are shown together with thecircle radius on the cutting tool coast side.



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Final part to be parameterized is the drive top, this is done by a modified super elliptic shape. The design domain is shown as

the hatched part in Fig. 4 and enlarged in Fig. 5. As seen in Fig. 4 the design domain size is variable and controlled through the

parameter μ , with the restrictions from the boundaries this parameter must fulfill.

μ min = −π

4+

5

4tanðαc Þ≦ μ ≦

π

4−

5

4tanðαdÞ = μ max ð10Þ

From the optimization presented in Ref. [1] it was found that in order to minimize the stress concentration it is important that

the parameterization includes a straight part before entering the elliptical shape, but in that paper the tooth were symmetric. The

idea used in the present paper is instead that the design domain can change size through the design parameter μ .

The remaining top part is as indicated in Fig. 5a parameterized by a super elliptical shape, only the first quarter of the super

ellipse is used. Parametric form of the super ellipse is

x = a0 + a1cosðt Þð2 =ηÞ

M ; t ∈½0 : 90

∘ ð11Þ

y = b0 + b1sinðt Þð2 =ηÞ

M ; t ∈½0 : 90

∘ ð12Þ

where the constants are given by

a0 = μ ; a1 =π

4−tanðαdÞ−μ

; b0 = 1; b1 =

1

4

As indicated in Fig. 5a the super ellipse might potentially come outside the design domain, which is not wanted since this has

an influence on the length of the involute of the cut tooth. To move the super ellipse back a distortion is added to the x position

parameterization. The distortion is indicated in Fig. 5b by rotating the dashed line. The quarter distorted super ellipse

parameterization is given by

x = a0 + a1cosðt Þð2 =ηÞ

1−b1

a1

tanðαdÞsinðt Þð2 =ηÞ

M ; t ∈½0 : 90

∘ ð13Þ

y = b0 + b1sinðt Þð2 =ηÞ

M ; t ∈½0 : 90

∘ ð14Þ

Using the parameterization given by Eqs. (13) and (14) it is possible both to achieve the design space upper limit by letting

η →∞ or the lower boundary by letting η →0. The given parameterization fulfills that the gradient/slope is continuous, i.e., no

jumps in the slope if η ≧1.

The presented total cutting tool tooth parameterization is in principle controlled by four parameters; the two pressure angles

α d and α c , the length parameter μ and the super elliptic power η . As previously stated one of the pressure angles is assumed given

so the optimizations presented in Sections 4 and 5 are parameter studies with only three parameters, μ , η and either α d or α c . It isshown that this simple parameterization is suf ficiently flexible to achieve constant stress along a major part of the root.

Fig. 5. Thetoothtop parameterization thatcuts thedriveside tooth root, here shown fora positive value of μ . a) Theparameterization of includinga super ellipse is

shown to be also outside the design domain. b) The super elliptical shape is forced back into the design domain by a distortion.



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4. Analytical description of the teeth shape

In shape optimization it is important to have a detailed or preferably analytical shape description. Analytical description also

makes verification and comparison possible for other designs. Another reason is that it is known from shape optimization (see e.g.

Ref. [23] and references therein) that we cannot use the nodes of the FE model as design parameters. In the present paper, we have

made an analytical cutting tool parameterization and it is possible to find analytical descriptions for the envelope of the

parameterizations in case of a gear with a finite number of teeth. This might not be as easy when using e.g. splines to parameterize

the tool tip.

In Ref. [1] it is shown as to how the envelope can be found for two different cutting tooth parameterizations. The

parameterization in the present paper is not identical and for this reason and for easy reference a method forfinding the envelope

analytically is presented.

4.1. Envelope of cutting tool

The cutting tooth is made by connecting curve segments. There are three different curves; distorted super ellipse, circle and

straight line. The envelope of these three curves is found separately although the circle is a special case of the distorted super

ellipse.

Gear teeth are cut by a simultaneous cutting tool movement and gear blank rotation, the same geometry is achieved by a

combined rotation and translation of the cutting tool. Rotation and translation are controlled by the angle (variable) θ.

All curves are described in parametric form, { f (t )·M , g(t )·M }T , rotating and translating this around the gear blank give the

following parameterization

xðt ;θÞ yðt ;θÞ

= M

cosðθÞ −sinðθÞsinðθÞ cosðθÞ

f ðt Þ−θ⋅ z = 2

g ðt Þ− z = 2− ps

; t ∈½0 : 90

∘: ð15Þ

where ps is a possible profile shift factor. Parameterization (18) is a whole set of curves controlled by the two parameters, t which

describes the position along the curve and θ that describes the curve translation and rotation.

The envelope of Eq. (18) is determined by differentiating Eq. (18) with respect to t and θ and demanding that these two vectors

are aligned, i.e., that

dxðt ;θÞ

dθ

dyðt ;θÞ

dt −

dyðt ;θÞ

dθ

dxðt ;θÞ

dt = 0 ð16Þ

By solving (19), θ(t ) is determined as a function of t . This function can then be put back into Eq. (18) to give the analytical

description of the envelope in parametric form.A straight segment between two points ( β 1M , γ 1M ) and ( β 2M , γ 2M ) has the parametric form.


= M


ð1−t Þβ1 + t β2−θ⋅ z = 2

ð1−t Þγ1 + t γ2− z = 2− ps

; t ∈½0 : 1 ð17Þ

Applying Eq. (19) for the straight segment we find

θðt Þ = 2ðβ1−β2Þ2

+ ðγ1−γ2Þ2

ðβ2−β1Þ zt + 2

β1ðβ1−β2Þ + ðγ1− psÞðγ1−γ2Þ

ðβ1−β2Þ z; t ∈½0 : 1 ð18Þ

The circle on the coast side has the parametric form.


= M


μ + κcosðt Þ−θ · z = 2

5 =4−κ + κsinðt Þ− z = 2− ps

; t ∈½90

∘: 180

∘−αc ð19Þ

Applying Eq. (19) for the circle we find

θðt Þ =4μ + 4ðκ + psÞ−5ð Þcot ðt Þ

2 z; t ∈½90

∘: 180

∘−αc ð20Þ

Parametric form for the distorted super ellipse can be given as

xðt ; θ yðt ; θÞ

= M


a0 + C ð2 =ηÞK −θ⋅ z = 2

b0 + b1S ð2 =ηÞ− z = 2− ps

( ); t ∈½0 : 90

∘ ð21Þ

where K =a1−b1 sin(t )(2/ η )tan(α d), C =cos(t ) and S =sin(t ).


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Applying Eq. (16) the result is not as simple as the previous examples but can be determined analytical as

θðt Þ = 2 b1C 2S ð2=ηÞð ps−b0−b1S ð2=ηÞÞ + a0b1C ð2 + 2=ηÞS ð2=ηÞtanðαdÞ + a0C ð2=ηÞS 2K + C ð4=ηÞS 2K 2 + b1C ð2 + 4 =ηÞS ð2=ηÞtanðαdÞK

= zC ð2=ηÞ

a1S 2 + b1S ð2=ηÞ

tanðαdÞðC 2−S 2Þ

; t ∈½0 : 90∘ ð22Þ

Finally if η b1 then the slope is no longer continuous and in this case we need the trajectory of a point. For a general point, ( β 1M ,

γ 1M ), the trajectory is given directly by

xðt Þ yðt Þ

= M

cosðt Þ −sinðt Þsinðt Þ cosðt Þ

β1−t · z = 2γ1− z = 2− ps

ð23Þ

The envelopes of the different cutting tool segments are now determined. Then bookkeeping must be applied tofind the overall

envelope of all the envelope parts. An example is given in Fig. 6.

The curves given in Fig. 6 are given in the specific case where η = 2, μ =0, α d=30° and α c =20° and the numberof teeth onthe

gear is z =17. The final tooth after the bookkeeping is presented in Fig. 7, also shown are the boundary conditions for the FE

analysis. The curves in Fig. 7 are all analytically determined but divided in segments as is the cutting tool that created them.

5. Optimization of general spur gear

As discussed in Ref. [1] the design that minimizes the maximum bending stress in the tooth will depend on the position of the

external loading on the tooth. Therefore in order to compare different designs the load must be applied in the same position. In

gear design an important quantity is the contact ratio, i.e., the average number of teeth in contact between two mating gears.

Normally there are either one or two gears in contact at the same time. In the present paper nothing is done to change the contact

ratioother than the influence that is related to the pressure angle at the drive side, i.e., all teeth are cut with a rack tooth that have a

height of 2.25 M . Loading on the tooth is at its highest value when there is only one tooth transmitting the torque. At the same time

Fig. 6. The envelope of the different curve segments and the trajectory of intersecting points. The final tooth is indicated by the gray area.

Fig. 7. The combined tooth from the curves in Fig. 6 with the used supports in the FE modeling indicated.




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the stress will be highest when the load is closest to the tip, the worst point is often referred to as the outer limit of single tooth

contact. The position of this point varies depending on both gears in the mesh. For consistency and for easy reference the choice

made here is, as in Ref. [1], that the tooth is loaded at the pitch point where we have rolling contact. The load is not put at a single

node but applied as a constant line load symmetrically around the pitch point.

In all examples presented in this section the starting point is the ISO tooth with α d= α c = 20°, the load size is scaled so that the

maximum of the largest principal stress is unity. To compare the bending stress of the optimized asymmetric teeth to the ISO tooth

the transferred torque is kept constant, i.e., the load size on the optimized tooth is scaled relative to the ISO tooth load.

First example is with 17 teeth, i.e., z =17, and we are at the limit of under-cutting. In Fig. 8 a plot with iso-lines and gray scale of

the largest principal stress is shown. The plot only shows the stress at points where the numerical largest principal stress is

positive, i.e., where there is predominating tension. If the numerical largest principal stress is negative, i.e., there is predominatingcompression, the color is white. In Fig. 8 the external loading on the tooth is also shown together with the reaction forces at the

clamped boundaries.

The load is scaled such that the maximum bending stress is unity, which is illustrated, by the scale in Fig.8. Fig. 9 shows a close-

up of the stress concentration zone of the ISO tooth. Fig. 9b shows the iso-lines of the largest positive principal stress as in Fig. 8 but

now without the gray scale. Fig. 9a shows the size of the largest principal stress along the part of the boundary where the stress

concentration is present. The stress size is indicated by the gray area, the perpendicular thickness of the gray area corresponds to

the stress level. A tensile stress is plotted under the boundary for illustrative purposes.

Fromthe stress plot in Fig. 9a it can be seen that there is a potential for improving the stress. However, the ISO tooth does have a

rather nice stress distribution along the boundary, so the room for improvement through only shape optimization is limited. This

was done in Ref. [1] where the best design for a gear with 17 teeth gave a stress reduction of 12.2% comparedto the ISO tooth. With

the asymmetric design we can also improve the stress by increasing the tooth root thickness.

Fig. 10 is similar to Fig. 8. The design is optimized through a parameter study. Fixed value of this optimization is the coast

pressure angle α c =20°.

Fig. 8. The ISO profile of a tooth for a gear with 17 teeth. Iso-lines and gray scale of the largest principal stress. In the plot the gray scale shows tension only while

compressive larger principal stresses are white.

Fig. 9. Close-up of the stress concentration zone for the ISO gear.




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The optimized design in Fig. 10 has the following design parameters α c =20°, α d=35°, η = 1.87 and μ = μ min. Stress is reduced

with 39.2% as compared to the ISO profile. In Fig. 11 a close-up of the interesting tooth part is given. Since this optimization is a

parameter study with only three design parameters there might be room for improvement with a more complicated boundary

parameterization. From Fig. 11a it is, however seen that the stress is constant over a long part of the boundary so any optimization

relative to this design must give minor changes. The stress scale in Figs. 9 and 11 is the same so the reduction in the stress level is

directly visualized.

The price wepay in this design relative to the original ISO tooth is a smaller contact ratio but this can befixed through a possible

longer tooth since the tooth top thickness is not near the limit. Alternatively to this wefix the drive side pressure angle at α d=20°

resulting in the same contact ratio as the ISO gear. Result of this optimization is presented in Figs. 12 and 13.

The optimized design in Fig. 12 has the following design parameters α c =34°, α d=20°, η =1.57 and μ = 0.07. Stress is reduced

with 23.2% as compared to the ISO profile. In Fig. 13 a close-up of the interesting tooth part is given.

Overall the improvements in the bending stresses are large. The largest stress improvement is possible with α dNα c , here we find

almost twice the improvement found when α dbα c . The improvement of 39.2% and 23.2% should be compared to the result in Ref. [1]

where thebestdesigngavea stress reductionof 12.2%, sothe influence from theenlargedtoothroot thickness is clear.Thechoiceof α dNα c is similar to what is found in e.g. Ref.[10] while α dbα c corresponds to thechoice in e.g. Ref. [14]. With the latter choice thecontact ratiois

constant but some articles report that the teeth becomes too stiff while with the first choice the contact ratio goes down. It is not the

present paper's intent to comment on these matters but instead focus directly on the possible bending stress improvements.

In Ref. [1] it was found that the size of the possible stress improvement depends on the number of teeth. The largest

improvements were found when the number of teeth is low. To examine if this is also the case for an asymmetric gear and for a

direct comparison with the results reported in Ref. [14] the next examples are with a gear with z =34. Optimization results are

presented in Fig. 14.

In Fig.14a the optimizedresultforfi

xedcoastsidepressureangle α c = 20°shows that the designparameters are α c =20°, α d=36°, η =1.8 and μ = μ min. Improvement in thebending stress as compared to the ISOprofile is41.2%.In Fig.14b theresult forfixeddriveside

Fig. 10. Full view of optimizedtooth. Thedesignedtooth is fora gear with17 teeth,the optimizeddesign variables are α c

=20°, α d

=35°, η =1.87and μ = μ min

. The

stress is reduced with 39.2% as compared to the ISO profile.

Fig. 11. Close-up of thestress concentration zone forthe optimizedgear(α c =20°, α d=35°, η =1.87 and μ = μ min). Thestress is reduced with39.2%as comparedto

the ISO profile.


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pressure angle α c = 20° shows that the design parameters are α c =34°, α d=20°, η =1.7 and μ =0.09. Improvement in the bending

stress as compared to theISO profile is 20.1%. In bothexamples the stress along the boundaries of the stress concentrationis relatively

constant indicating thatan optimal or close to optimal result is obtained with this simple cuttingtool parameterization. In Ref. [14] the

reported improvement in the bending stress is 17% which can be compared directly with the 20.1% found here. Close agreement is

found although the results in [14] were found without using shape optimization. It is also noticed that in contradiction to the results

from optimizing only theroot shape of thestandard tooththe results are better for a gear with more teeth when α dNα c . However, the

Fig. 13. Close-up of the stress concentration zone for the optimized gear (α c =34°, α d=20°, η =1.57 and μ =0.07). The stress is reduced with 23.2% as compared

to the ISO profile.

Fig. 14. a) Close-up of thestress concentration zone forthe optimized gear with34 teeth, thedesign parameters shown are α c =20°, α d=36°, η =1.8 and μ = μ min.

The stress is reduced with 41.2% as compared to the ISO profile. b) Close-up of the stress concentration zone for the optimized gear with 34 teeth, the designparameters shown are α c =34°, α d=20°, η =1.7 and μ =0.09. The stress is reduced with 20.1% as compared to the ISO profile.

Fig. 12. Full view of optimizedtooth. Thedesignedtoothis fora gear with 17 teeth,the designparameters areα c

=34°,α d

=20°, η

=1.57and μ

=0.07. Thestress is

reduced with 23.2% as compared to the ISO profile.





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improvement in the bending stress is of the same order as with z=17. To verify this we finally optimize a gear with z =68.

Optimization results are presented in Fig. 15.

Results are of the same order for this gear. Improvements in the bending stress found are 44.3% and 19.4%, respectively.

From the performed parameter studies it is found that the reduction in the bending stress is not very sensitive to small changes

in the design parameters. This leads to the idea of a standard or two standard cutting racks, these are presented in the next section.

6. Suggested new standard asymmetric rack cutter

From the optimized designs presented in the previous section, specifically the design parameter values, it seems that it is

possible to make two standard rack cutters, one where the drive side pressure angle is fixed at α d=20° and another where the

coast side pressure angle is fixed at α c =20°. The suggested design variables, for the cutters, are given in Table 1.

Cutter designs are shown in Figs. 16 and 17.

Using a standard cutter for all gears, i.e. a non custom cutter, does result in designs that are not fully optimized. The differences

are, however, not large as seen in Fig. 18.

From the results in Fig. 18 the following conclusions can be made.

• The largest reduction in the bending stress can be found with α dNα c .

• With a drive side pressure angle,α d

=36° (AE1), the bending stress reduction compared to the standard ISO tooth is about 40%

independent of the number of teeth on the gear. Maximum difference as compared to an optimized tooth is 3%.

• With a coast side pressure angle, α c =34° (AE2), the bending stress reduction compared to the standard ISO tooth is about 18%

independent of the number of teeth on the gear. Maximum difference as compared to an optimized tooth is 5%.

As expected theperformance of the suggestedtwo newstandardrack cutters is notas excellent as thespecific optimizedcutters for

a gear with a given number of teeth. However, the difference is not large and thereduction in the bending stress compared to the ISO

tooth is still significant. A decision whether to use a standard cutter or a custom cutter must be problem dependent.

7. Conclusion

Results presented in the present paper show that large improvements in the bending stress for gears can be found by the use of

asymmetric gears. Bending stress reduction is achieved by two contributions, a thicker tooth root and a root shape change where

we have the stress concentration. The factor that has the largest influence here is the enlargement of the root thickness.

Optimization has been exemplified by three gears with the number of teeth being z =17, z =34 and z =68, respectively.

The cutting tool is designed so the root shape optimization of the gear tooth is achieved in an indirect way. However the

changes made to the cutting tool are directly related to the actual gear tooth. The design parameter choice of the optimization has

been that either the coast side pressure angle α c or the drive side pressure angle α d is fixed at 20°. Maximum reported reduction in

the bending stress is 44.3%, in this case we have a custom cutting tool specifically for a gear with 68 teeth. The paper proposes the

Fig. 15. a) Close-up of the stress concentration zone for the optimized gear with 68 teeth, the design parameters shown are α c =20°, α d=36°, η =1.94 and

μ = μ min. The stress is reduced with 44.3% as compared to the ISO profile. b) Close-up of the stress concentration zone for the optimized gear with 68 teeth, the

design parameters shown are α c =35°, α d=20°, η =1.81 and μ = 0.1. The stress is reduced with 19.4% as compared to the ISO profile.

Table 1

Design variables for suggested standard asymmetric elliptic cutters.

Asymmetric elliptic cutter α d α c η μ

AE 1 36° 20° 2 μ min = −π

4+

5

4tanð20∘Þ

AE 2 20° 34° 2 μ min = −

π

4 +5

4tanð34∘

Þ



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use of a standard rack or two standard racks. Reduction in the bending stress is lower than that reported for specific optimizations,

but the difference is not significant, especially for gears with a higher number of teeth.

Increasing the pressure angle on the drive side also leads to some negative effects that must be considered as discussed in

Section 2, and in the literature. The focus of the present paper is entirely on the bending stress. In a practical application these

effects should however be included.

Overall the paper has demonstrated that with a simple rack cutter parameterization with only four parameters (only three are

used actively in the presentedoptimizations) we can reducethe bending stressrather significantly. With a high drive side pressure

angle the bending stress improvements is in the order of 40% independent of the number of teeth on the gear, with a high coast

side pressure angle the improvement is roughly half the size. This also holds for the two new suggested standard rack cutters.

Acknowledgment

For the discussions and suggestions I wish to thank Prof. Pauli Pedersen.

Fig. 16. New standard asymmetric elliptic cuter AE1.

Fig. 17. New standard asymmetric elliptic cuter AE2.

Fig. 18. The reduction in percent of the bending stress, σ b, for the new standard cutter and the optimized cutters relative to the standard ISO teeth. For the number

of teeth equal to z =17, z =34 and z =68 respectively.





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